Help Regarding Z transform and DC gain

[moonnightingale]

For many days i am stuck with 2 queries. One is for DC gain and other for Z transform.

1. A question says to find DC gain of both G(s) and G(z) . G(s) is transfer function in continuous domain and G(z) in discrete domain.

G(s)= 10(s+1)/(s^2+s+10)

I have calculated G(z) by Matlab and by hand

clc
s=tf('s');
G=10*(s+1)/(s^2+s+10)
T=0.01;
G_d=c2d(G,T,'zoh')
grid on
pzmap(G_d);

What he mean by DC gain. How to calculate that in Matlab

2. Second question is regarding Z transform. T. How to calculate z transform of term which is having exp

for example Z transform of [exp^(st/2) ] / s^2 or [exp^(-1.5Ts)] / [(s+1)(s+3)] how to solve such questions. Kindly refer some book b/c ogata book does not explain it. ( s is laplace domain)

Thanks a lot for ur help

 

[dedalus]

1) DC gain is value of transfer function G at "zero frequency", so in continuous domain DC gain = G(s=0) and in discrete domain DC gain = G(z=1).

2) I suppose "st" and "Ts" in your examples are constants, because transfer function of linear system can be only polynomial function of "s". In this case you can just carry exponential term outside the transform: Z{[exp^(st/2) ] / s^2} = [exp^(st/2) ] *Z{1/ s^2}

 

[moonnightingale]

Thanks for first reply

Regarding second question s is laplace Domain
WE cannot take it out of equation